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For example, when using Sulfuric Vortex does its owner need to remind each player at the beginning of their upkeep to take the damage? Or is a case of, if its owner forgets to tell you, then you do nothing?

Similar question for Falkenrath Noble: if I forget to gain a life and forget to target a player to lose a life then does the ability never take effect? When is it too late - the next player's upkeep? The end of my turn?

At the beginning of each player's upkeep, Sulfuric Vortex deals 2
damage to that player.

It says "deals damage" instead of "you may have Sulfiric Vortex deal 2 damage". Without the "may", it's not optional. The Infraction Procedure Guide says:

You aren't allowed to miss your own triggered abilities, and your
opponent isn't required to remind you about them.

So not announcing the trigger is a rules infraction. If you do forget a trigger in a casual game (regular rule enforcement):

If a triggered ability is forgotten (or any other game error is made)
and the error is discovered later, players should alert the judge. The
judge will try and fix the error if possible and, if not, the game
will just continue.

The responsibility for announcing the trigger lies with the player who owns the ability. While other players are not required to announce the trigger, it's a good idea to do so. If the owner notices two rounds later, it will be applied retroactively.

Official tournaments are different (competitive and professional rule enforcement):

What the judge will do when you miss your triggers:
your opponent gets to decide whether he or she wants the trigger to
happen or not.

Looks like the rules just changed! "On a brighter note, it is never an infraction to not mention a triggered ability you don't control. This change was made several months ago"... I'm updating the answer. :)
–
AndomarMay 30 '13 at 15:15

lol thanks for looking into this further! My nephew had Sulfuric Vortex but kept forgetting to remind me about the trigger, so I wasn't sure if I should not apply it or keep reminding him at the start of my turn.
–
DamainmanMay 30 '13 at 15:19

the crux of current rules is basically "you should remember your own abilities, if you forget, and remember later, your opponent gets to choose if it happens". for casual play however you might not want to enforce that, losing due to that can leave a sour taste
–
PattersMay 30 '13 at 15:21

It's important to note that all of these are floor rules - they're not part of the rules of the game, but part of the tournament structure around the game. If these are games you're playing with your nephew, I would encourage you to continually remind him but also to devise some way of making sure that he remembers - for instance, putting a die on top of his library so he can't draw a card without looking around to see if there's anything that would be triggering in his upkeep.
–
Steven StadnickiMay 30 '13 at 16:15

This is going to cover competitive REL environments. at regular REL, it is possible for the judge to fix the game-state after triggers have been missed.

The trigger rules have been in a state of flux over the past couple years.

2 years ago, if your opponent missed a mandatory trigger and it was to your advantage and it was determined that you knew about it, purposefully let it go so that it was to your advantage, than YOU would get disqualified for cheating.

If it was determined to not be on purpose than you both would receive a warning

last year, Andomar's answer would be correct, but they changed that some months ago.

you can read the article for more details, but essentially the trigger is considered to not be missed until the controller of the trigger's source makes a decision where the trigger would have mattered.

Under the previous rules, if you didn't explicitly declare the trigger, than you would have considered to have missed it, but under today's rules, if your opponent tries to block with just 1 creature, you can correct them and remind them of the wolf's trigger. If you don't remind them it is considered missed